# Idea

The IPC-property (“inductive limit product commutation property”) is a technical condition on a category $A$ which ensures that presheaves with values in $A$ have a good notion of sheafification.

# Definition

A category $A$ is said to satisfy the IPC-property if

• for

• every family $\{I_s\}_{s \in S}$ of small filtered categories

• and for any family $\{\alpha_s : I_s \to A\}$ of functors

• indexed by a small set $S$

the canonical morphism

$colim \left( \prod_s I_s \stackrel{k \mapsto \prod_s \alpha_s(\pi_s(j))}{\to} A \right) \to \prod_s colim \alpha_s$

is an isomorphism.

# References

The IPC property is definition 3.1.10 in

It is invoked for sheafification in section 17.4 there.

Last revised on November 1, 2017 at 19:08:10. See the history of this page for a list of all contributions to it.